Linear subspace
A subspace, part vector space, linear subspace or linear subspace is in mathematics a subset of a vector space, which itself again is a vector space. The vector space operations vector addition and scalar multiplication of the output space are inherited to the sub-vector space. Every vector space contains itself and the zero vector space as trivial subspaces.
Each subspace is the product of linearly independent subset of vectors of the output space. The sum and the intersection of two subspaces yields again a sub-vector space whose dimension can be determined by the dimension formula. Each sub- vector space has at least one complementary space, so that the output space is the direct sum of the subspace and its complement. Further, each sub- vector space a factor space are allocated, resulting from the fact that all the elements of the output space along the subspace are projected parallel.
Subspaces are used in linear algebra, among other things, to characterize core and image of linear transformations, the solution sets of linear equations and eigenspaces of eigenvalue problems. In the functional analysis, in particular subspaces of Hilbert spaces, Banach spaces and dual spaces are investigated. Subspaces have a variety of applications, such as numerical methods for solving large systems of linear equations and for partial differential equations with optimization problems in coding theory and signal processing.
- 3.1 vector space axioms
- 3.2 Description
- 4.1 intersection and union
- 4.2 sum
- 4.3 Direct sum
- 4.4 Multiple operands
- 5.1 Complementary space
- 5.2 factor space
- 5.3 Annihilatorraum
- 6.1 Linear maps
- 6.2 Linear Equations
- 6.3 Eigenvalue Problems
- 6.4 Invariant subspaces
- 7.1 Hilbert spaces
- 7.2 Unterbanachräume
- 7.3 Topological dual spaces
Definition
Is a vector space over a field, then a subset is then exactly one subspace of if it is not empty and closed under vector addition and scalar multiplication. So there must
Apply to all vectors and all scalars. The vector addition and scalar multiplication in the subspace are the restrictions of the corresponding operations of the output space. Equivalent to the first condition, you can also request that the zero vector of is included.
Using these three criteria can be used to check whether a given subset of a vector space also forms a vector space, without having to prove all the vector space axioms. A subspace is often referred to as " subspace " when the context is clear that this is a linear subspace and not a general subspace.
Examples
Concrete examples
The set of all vectors of the real number plane forms with the usual component-wise vector addition and scalar multiplication of a vector space. The subset of vectors that applies to the, forms a subspace of, because it applies to all:
- The origin is in
As another example, one can consider the vector space of all real functions with the usual pointwise addition and scalar multiplication. In this vector space, the set of linear functions form a subspace, because it applies to:
- The zero function is
- , thus
- , thus
More general examples
- Every vector space contains itself and the zero vector space, which consists only of the zero vector, as trivial subspaces.
- In the vector space of real numbers are the amount and all the single subspaces.
- In the vector space of complex numbers are the set of real numbers and the set of imaginary numbers subspaces.
- In the Euclidean plane all lines form through the zero point subspaces.
- In Euclidean space all lines through the origin and origin planes forming subspaces.
- In the vector space of all polynomials of the set of polynomials of maximum degree forms a subspace for each natural number.
- In the vector space of square matrices, the symmetric and skew-symmetric matrices form subspaces respectively.
- In the vector space of real functions on an interval, the integrable functions, continuous functions and differentiable functions form subspaces respectively.
- In the vector space of all maps between two vector spaces over the same body, the amount of linear maps forms a subspace.
Properties
Vector space axioms
The three subspace criteria are actually necessary and sufficient for the validity of all the vector space axioms. Due to the seclusion of the tonnage is namely for all vectors by setting
And thus further by setting
This includes the amount in particular the zero vector, and each element and the additive inverse element. So is a subset of and thus in particular an abelian group. The associative law, commutative, distributive laws and the neutrality of the fuel transferred to directly from the output space. This fulfills all the vector space axioms and is also a vector space. Conversely, any subspace must satisfy the three criteria specified as the vector addition and scalar multiplication are the limitations of the corresponding operations.
Representation
Every subset of vectors of a vector space spanned by the formation of all possible linear combinations
A sub- vector space of which is called the linear hull of. The linear hull is the smallest subspace that includes the amount and equal to the average of all subspaces of which include. Conversely, any subspace is the product of a subset of, ie, it applies
Where the amount is called a system of generators. A minimal generating system consists of linearly independent vectors and is called a basis of a vector space. The number of elements of a base represents the dimension of a vector space.
Operations
Intersection and union
The intersection of two subspaces of a vector space
Is always itself a subspace. The union of two subspaces
However, is only a subspace if or applies. Otherwise, the association is indeed closed under scalar multiplication, but not with respect to the vector addition.
Sum
The sum of two subspaces of a vector space
Is again a subspace, namely the smallest subspace that contains. For the sum of two finite-dimensional vector spaces the dimension formula holds
Resulting in reversed also the dimension of the intersection of two subspaces can be read. Section and total bases of vector subspaces of finite dimension can be calculated with the Zassenhaus algorithm.
Direct sum
The intersection of two subspaces consists only of the zero vector, then, is so refers to the sum as an internal direct sum
Since it is isomorphic to the external direct sum of two vector spaces. In this case, for every vectors uniquely determined, with. From the dimension theorem then follows because the zero vector space is zero-dimensional, the dimension of the direct sum
What is true in the infinite-dimensional case.
More operands
The previous operations can be generalized also to more than two operands. Is a family of subspaces of, with an arbitrary index set, then forming the average of these subspaces
Again a subspace of. Next also yields the sum of several subspaces
Again a subspace of, and in the case of an index set with infinitely many elements only finitely many summands must be equal to the zero vector. Such a sum is called directly and then with
Referred to as the cut each subspace yields the zero vector space with the sum of the remaining subspaces. This is equivalent to the fact that each vector has a unique representation of the sum of elements of the vector spaces.
Derived rooms
Complementary space
For each subspace of is at least one complementary space so that
Applies. Each such complementary space corresponds exactly to a projection onto the subspace, ie an idempotent linear map, with the
Holds, where the identity map is. In general, several complementary spaces exist for a vector subspace of which by the vector space structure is not excellent. In Skalarprodukträumen however, it is possible to speak of mutually orthogonal subspaces. Is finite, then there exists for every vector subspace a uniquely determined orthogonal complementary space, which is just the orthogonal complement of, and it is then
Factor space
Each subspace of a vector space can be a factor space are allocated, resulting from the fact that all elements of the subspace are identified with each other and so the elements of the vector space along the subspace are projected parallel. Formally, the space factor is defined as the set of equivalence classes
Of vectors in, where the equivalence class of a vector
The amount of the vectors is different from only one element of the vector space. The factor space forms a vector space, if the vector space operations are defined as representatives, but it is not itself a subspace of. For the dimension of the factor space
The subspaces of are precisely the factor spaces, where subspace of with is.
Annihilatorraum
The dual space of a vector space over a field is the space of linear maps from to and thus itself a vector space. For a subset of the set of all functionals that vanish on a subspace of the dual space, the so-called Annihilatorraum
Is finite, then for the dimension of the Annihilatorraums a subspace of
The dual space of a subspace is thus isomorphic to the factor space.
Subspaces in linear algebra
Linear maps
Is a linear map between two vector spaces and the same body, then forms the core of the figure
A subspace of the image and the image
A subspace of. Furthermore, the graph of a linear map is a subspace of the product space. If the vector space is finite, then for the dimensions of the involved spaces of rank theorem
The dimension of the image is also called rank and the dimension of the core and holes of the linear mapping. After the homomorphism the image is isomorphic to the factor space.
Linear Equations
In turn, is a linear map between two vector spaces over the same body, then the solution set of the homogeneous linear equation
A subspace of, precisely the core of. The amount of solution of an inhomogeneous linear equation
With, however, is if it is not empty, an affine- linear subspace of, which is a consequence of the superposition property. The dimension of the solution space is then also equal to the dimension of the core of.
Eigenvalue problems
Is now a linear transformation of a vector space into itself, ie an endomorphism, with associated eigenvalue problem
Then each is related to an eigenvalue eigenspace
A subspace of whose different from the zero vector elements are exactly the corresponding eigenvectors. The dimension of the eigenspace corresponding to the geometric multiplicity of the eigenvalue; it is at most as large as the algebraic multiplicity of the eigenvalue.
Invariant subspaces
Is again an endomorphism, then is called a subspace of invariant or short - invariant if
Applies, that is, when all the image is also in. The image below, then, is a subspace of. The trivial subspaces and, as well, and all eigenspaces of are always under invariant. Another important example of invariant subspaces are the main rooms, which are used for example in determining the jordan between normal form.
Subspaces in the Functional Analysis
In Hilbert spaces
In Hilbert spaces, ie complete Skalarprodukträumen, in particular sub- Hilbert spaces are considered, ie subspaces that the restriction of the scalar product with respect to still incomplete. This property is equivalent to saying that the vector subspace closed under the norm topology induced by the scalar product is. Not every subspace of a Hilbert space is also complete, it can, however, to each incomplete subspace by the conclusion of training eligible for a Hilbert space in which that is then sealed. For each sub- Hilbert space exists after the projection set and a uniquely determined orthogonal complement, which is always complete.
In Hilbert spaces play an important role in quantum mechanics and the Fourier - or multi-scale analysis of signals.
Unterbanachräume
In Banach spaces, ie complete normed spaces, one can analogously Unterbanachräume, ie subspaces that the restriction of the standard are complete with respect to, consider. As in the case of Hilbert space is a subspace of a Banach space if and only a Unterbanachraum when it is completed. Next can be to each incomplete subspace of a Banach space obtained by completing a Unterbanachraum which is close to this. However, there is no complementary Unterbanachraum generally to a Unterbanachraum.
In a semi- normed space, the vectors with zero seminorm form a subspace. For half a normed space normed space obtained as the factor space by the equivalence classes of vectors, which do not differ with respect to the semi-norm is considered. Is the semi- normed space completely, so this factor space is then a Banach space. This construction is especially used in the Lp- spaces and related function spaces.
In the numerical calculation of partial differential equations using the finite element method, the solution is approximated in a suitable finite Unterbanachräumen the underlying Sobolevraums.
Topological dual spaces
In the functional analysis is considered in addition to the algebraic dual space and the topological dual space of a vector space, which consists of the continuous linear maps from to. For a topological vector space of topological dual space forms a vector subspace of the algebraic dual space. By the theorem of Hahn- Banach has a linear functional on a subspace of a real or complex vector space, which is limited by a sublinear function, a linear continuation on the total space, which is also limited by these sub-linear function. As a consequence of the topological dual space of a normed space contains a sufficient number of functionals, which forms the basis of a rich duality theory.
Other applications
Other important applications of vector subspaces are:
- The Gram -Schmidt orthogonalization to construct Orthogonalbasen
- Krylov subspace methods for solving large sparse systems of linear equations
- Solution methods for optimization problems
- Linear codes in coding theory